Stochastic Growth Model

KoenVermeylen TheStochasticGrowthModel Downloadfreebooksat Download free eBooks at bookboon.com Click on the ad to read more The Stochastic Growth Model 2 Contents 1. Introduction 2. The stochastic growth model 3. The steady state 4. Linearization around the balanced growth path 5. Solution of the linearized model 6. Impulse response functions 7. Conclusions Appendix A A1. The maximization problem of the representative fi rm A2. The maximization problem of the representative household Appendix B Appendix C C1. The linearized production function C2. The linearized law of motion of the capital stock C3. The linearized fi rst-order condotion for the fi rm’s labor demand C4. The linearized fi rst-order condotion for the fi rm’s capital demand C5. The linearized Euler equation of the representative household C6. The linearized equillibrium condition in the goods market References Contents 3 4 7 8 9 13 18 20 20 20 22 24 24 25 26 26 28 30 32 Designed for high-achieving graduates across all disciplines, London Business School’s Masters in Management provides specific and tangible foundations for a successful career in business. This 12-month, full-time programme is a business qualification with impact. In 2010, our MiM employment rate was 95% within 3 months of graduation*; the majority of graduates choosing to work in consulting or financial services. As well as a renowned qualification from a world-class business school, you also gain access to the School’s network of more than 34,000 global alumni – a community that offers support and opportunities throughout your career. For more information visit www.london.edu/mm, email mim@london.edu or give us a call on +44 (0)20 7000 7573. Masters in Management The next step for top-performing graduates * Figures taken from London Business School’s Masters in Management 2010 employment report Download free eBooks at bookboon.com The Stochastic Growth Model 3 Introduction 1. Introduction This article present s the stoch a stic growth model. The stochastic growth model is a stochastic version of the neoclassical growth model with microfoundations, 1 and provides the backbone of a lot of macroeconomic models that are used in modern macroeconomic research. The most popular way t o solve the stochastic growth model, is to l inearize the model around a steady state, 2 and to solve the linearized model with the method of undetermined coefficients . This solution method is due to Campbell (1994). The set-up of the stochastic growth mode l is given in the next section. Section 3 solv e s for the steady s tate, around w hich the model is linearized in section 4. The linearized model is then solved in section 5. Section 6 show s how the economy responds to stochastic shocks. Some c oncluding remarks are given i n section 7. Download free eBooks at bookboon.com The Stochastic Growth Model 4 The representative firm Assume that the production side of the economy is represented by a representative firm, which produces output ac c ording to a Cobb-Douglas production function: Y t = K α t (A t L t ) 1−α with 0 <α<1(1) Y is aggregate output, K is the aggregate capital stock, L is aggregate labor supply and A is a technology parameter. The subscript t denotes the time period. The aggregate capital stock depends on aggregate investment I and the depreci- ation rate δ: K t+1 =(1− δ)K t + I t with 0 ≤ δ ≤ 1(2) 2. The stochastic growth model The productivity parameter A follows a stochastic path with trend growth g and an AR(1) stochastic component: ln A t =lnA ∗ t + ˆ A t ˆ A t = φ A ˆ A t−1 + ε A,t with |φ A | < 1(3) A ∗ t = A ∗ t−1 (1 + g) The stoch astic shock ε A,t is i.i.d. with mean zero. The goods market alway s clears, such that the firm alway s sells i ts total pro- duction. Taking current and future factor prices as given, the firm hires labor and invests in its capital stock to maximize its current value. This leads to the following first-order-conditions: 3 (1 − α) Y t L t = w t (4) 1=E t  1 1+r t+1 α Y t+1 K t+1  + E t  1 − δ 1+r t+1  (5) According to equation (4), the firm hires labor until the marginal product of labor is equal to its m arginal c ost (whic h is the real wage w). Equation (5) s hows that the firm’s investment demand at time t is such that the marginal cost of in vestment, 1, is equal to the expected discoun te d marginal product o f capital at time t + 1 plus the expected discounted value of the extra capital stock which is left after depreciation at time t +1. The stochastic growth model Download free eBooks at bookboon.com Click on the ad to read more The Stochastic Growth Model 5 The government The government consumes every period t an amount G t , which follow s a stochastic path with trend growth g and an AR(1) stochastic component: ln G t =lnG ∗ t + ˆ G t ˆ G t = φ G ˆ G t−1 + ε G,t with |φ G | < 1(6) G ∗ t = G ∗ t−1 (1 + g) The stochastic shock ε G,t is i.i.d. with mean zero. ε A and ε G are uncorrelated at all leads and lags. The government finances its consumption by issuing public debt, subject to a transversality condition, 4 and by raising lump-sum taxes. 5 The timing of taxation is irrelevant because of Ricardian Equivalence. 6 The stochastic growth model “The perfect start of a successful, international career.” CLICK HERE to discover why both socially and academically the University of Groningen is one of the best places for a student to be www.rug.nl/feb/education Excellent Economics and Business programmes at: Download free eBooks at bookboon.com The Stochastic Growth Model 6 The representative household There is one represen tative household, who derives utility from her current and future consumption: U t = E t  ∞  s=t  1 1+ρ  s−t ln C s  with ρ>0(7) The parameter ρ is c alled the s ubjective discount rate. Every period s, the household starts off with her assets X s and receives interest payments X s r s . She also supplies L units of labor to the representative firm, and therefore receives labor income w s L. Tax pa yments are lump-sum and amount to T s . She then decides how much she consumes, and how much assets she will hold in her portfolio unt il period s + 1. This leads to her dynamic budget constraint: X s+1 = X s (1 + r s )+w s L − T s − C s (8) We need to make sure that the household does not incur ever increasing debts, which she will never be able to pay back anymore. Under plausible assumptions, this implies that over an infinitely long horizon the present discounted value of the household’s assets must be zero: lim s→∞ E t  s  s ′ =t 1 1+r s ′  X s+1  =0 (9) This equation is called the transversality condition. The household then takes X t and the current and e xpected values of r, w,andT as giv e n, and chooses her consumption path to maximize her utilit y (7) subject to her dynamic budget constraint (8) and the transversalit y condition (9). This leads to the following Euler equation: 7 1 C s = E s  1+r s+1 1+ρ 1 C s+1  (10) Equilibrium Every period, the factor markets and the goods market clear. For the labor market, we already implicitly assumed this by using the same notation (L) for the representative h ousehold’s labor supply and the representative firm’s labor demand. Equilibrium in the goods market requires that Y t = C t + I t + G t (11) Equilibrium in the capital market follows then from Walras’ law. The stochastic growth model Download free eBooks at bookboon.com Click on the ad to read more The Stochastic Growth Model 7 Let us now d erive the model’s balanced growth path (or steady state); variables evaluated on the balanced growth path are denoted by a ∗ . To derive the balanced gro wth path, we assume that b y sheer luc k ε A,t = ˆ A t = ε G,t = ˆ G t =0,∀t. The model then becomes a standard neo classical growth model, for w hich the solution is given by: 8 Y ∗ t =  α r ∗ + δ  α 1−α A ∗ t L (12) K ∗ t =  α r ∗ + δ  1 1−α A ∗ t L (13) I ∗ t =(g + δ)  α r ∗ + δ  1 1−α A ∗ t L (14) C ∗ t =  1 − (g + δ) α r ∗ + δ  α r ∗ + δ  α 1−α A ∗ t L − G ∗ t (15) w ∗ t =(1− α)  α r ∗ + δ  α 1−α A ∗ t (16) r ∗ =(1+ρ)(1 + g) − 1 (17) 3. The steady state The steady state © Agilent Technologies, Inc. 2012 u.s. 1-800-829-4444 canada: 1-877-894-4414 Teach with the Best. Learn with the Best. Agilent offers a wide variety of affordable, industry-leading electronic test equipment as well as knowledge-rich, on-line resources —for professors and students. We have 100’s of comprehensive web-based teaching tools, lab experiments, application notes, brochures, DVDs/ CDs, posters, and more. See what Agilent can do for you. www.agilent.com/find/EDUstudents www.agilent.com/find/EDUeducators Download free eBooks at bookboon.com The Stochastic Growth Model 8 Let us now linearize t he model presented in section 2 around the balance d grow th path derived in section 3. Loglinear deviations from the balanced growth path are denoted b y aˆ(so that ˆ X =lnX − ln X ∗ ). Below a re the loglinearized versions of the production function ( 1), the law of motion of the capital stock (2), the first-order conditions (4) and (5), the Euler equation (10) and the equilibrium condition ( 11): 9 ˆ Y t = α ˆ K t +(1− α) ˆ A t (18) ˆ K t+1 = 1 − δ 1+g ˆ K t + g + δ 1+g ˆ I t (19) ˆ Y t =ˆw t (20) E t  r t+1 − r ∗ 1+r ∗  = r ∗ + δ 1+r ∗  E t ( ˆ Y t+1 ) − E t ( ˆ K t+1 )  (21) 4. Linearization around the balanced growth path ˆ C t = E t  ˆ C t+1  − E t  r t+1 − r ∗ 1+r ∗  (22) ˆ Y t = C ∗ t Y ∗ t ˆ C t + I ∗ t Y ∗ t ˆ I t + G ∗ t Y ∗ t ˆ G t (23) The loglinearized la ws of motion of A and G are given by equations (3) and (6): ˆ A t+1 = φ A ˆ A t + ε A,t+1 (24) ˆ G t+1 = φ G ˆ G t + ε G,t+1 (25) Linearization around the balanced growth path Download free eBooks at bookboon.com The Stochastic Growth Model 9 I now solve the linearized model, which is described by equations (18) until (25). First note that ˆ K t , ˆ A t and ˆ G t are known in the beginning of period t: ˆ K t dep ends on past investment decisions, and ˆ A t and ˆ G t are determined by current and past values of respectively ε A and ε G (which are exogenous). ˆ K t , ˆ A t and ˆ G t are therefore called period t’s state variables. The values of the other variables in period t are endogenous, however: investment and consumption are chosen by the representative firm and the representative household in such a way that they maximize their profits and utility ( ˆ I t and ˆ C t are therefore called period t’s control variables); the values of the interest rate and the wage are such that they clear the capital and the labor marke t. Solving the model requires that we express period t’s endogenous variables as functions of period t’s state variables. The solution of ˆ C t , f or instance, therefore looks as follows: ˆ C t = ϕ CK ˆ K t + ϕ CA ˆ A t + ϕ CG ˆ G t (26) The cha llenge now is to determine the ϕ-coefficients. First substitute equation (26) in t he Euler equation (22): ϕ CK ˆ K t + ϕ CA ˆ A t + ϕ CG ˆ G t = E t  ϕ CK ˆ K t+1 + ϕ CA ˆ A t+1 + ϕ CG ˆ G t+1  − E t  r t+1 − r ∗ 1+r ∗  (27) Now eliminate E t [(r t+1 −r ∗ )/(1+r ∗ )] with equation (21), and use equations (18), (24) and (25) to eliminate ˆ Y t+1 , ˆ A t+1 and ˆ G t+1 in the resulting expression. This 5. Solution of the linearized model Solution of the linearized model Download free eBooks at bookboon.com The Stochastic Growth Model 10 leads to a relation between period t’s state variables, the ϕ-coefficients and ˆ K t+1 : ϕ CK ˆ K t + ϕ CA ˆ A t + ϕ CG ˆ G t =  ϕ CK +(1− α) r ∗ + δ 1+r ∗  ˆ K t+1 +  ϕ CA − (1 − α) r ∗ + δ 1+r ∗  φ A ˆ A t + ϕ CG φ G ˆ G t (28) We now derive a second r elation be tween period t’s state variables, the ϕ-coe fficients and ˆ K t+1 : rewrite the law of motion (19) by eliminating ˆ I t with equation (23); eliminate ˆ Y t and ˆ C t in the resulting equation with the production function (18) and expression (26); note that I ∗ = K ∗ (g + δ); and note that (1 − δ)/(1 + g)+ (αY ∗ t )/(K ∗ t (1 + g)) = (1 + r ∗ )/(1 + g). This yields: ˆ K t+1 =  1+r ∗ 1+g − C ∗ K ∗ (1 + g) ϕ CK  ˆ K t +  (1 − α)Y ∗ K ∗ (1 + g) − C ∗ K ∗ (1 + g) ϕ CA  ˆ A t −  G ∗ K ∗ (1 + g) + C ∗ K ∗ (1 + g) ϕ CG  ˆ G t (29) Substituting equation (29) in equation (28) to eliminate ˆ K t+1 yields: ϕ CK ˆ K t + ϕ CA ˆ A t + ϕ CG ˆ G t =  ϕ CK +(1− α) r ∗ + δ 1+r ∗  1+r ∗ 1+g − C ∗ K ∗ (1 + g) ϕ CK  ˆ K t +  ϕ CK +(1− α) r ∗ + δ 1+r ∗  (1 − α)Y ∗ K ∗ (1 + g) − C ∗ K ∗ (1 + g) ϕ CA  ˆ A t −  ϕ CK +(1− α) r ∗ + δ 1+r ∗  G ∗ K ∗ (1 + g) + C ∗ K ∗ (1 + g) ϕ CG  ˆ G t +  ϕ CA − (1 − α) r ∗ + δ 1+r ∗  φ A ˆ A t − ϕ CG φ G ˆ G t (30) As this equation must hold for all values of ˆ K t , ˆ A t and ˆ G t , we find the following system of three equations and three unknowns: ϕ CK =  ϕ CK +(1− α) r ∗ + δ 1+r ∗  1+r ∗ 1+g − C ∗ K ∗ (1 + g) ϕ CK  (31) ϕ CA =  ϕ CK +(1− α) r ∗ + δ 1+r ∗  (1 − α)Y ∗ K ∗ (1 + g) − C ∗ K ∗ (1 + g) ϕ CA  +  ϕ CA − (1 − α) r ∗ + δ 1+r ∗  φ A (32) ϕ CG = −  ϕ CK +(1− α) r ∗ + δ 1+r ∗  G ∗ K ∗ (1 + g) + C ∗ K ∗ (1 + g) ϕ CG  − ϕ CG φ G (33) Solution of the linearized model [...]... variables converge back to their steady state values Download free eBooks at bookboon.com 17 Conclusions The Stochastic Growth Model 7 Conclusions This note presented the stochastic growth model, and solved the model by first linearizing it around a steady state and by then solving the linearized model with the method of undetermined coefficients You’re full of energy and ideas And that’s just what we are... The Stochastic Growth Model 1 Microfoundations means that the objectives of the economic agents are formulated explicitly, and that their behavior is derived by assuming that they always try to achieve their objectives as well as they can 2 A steady state is a condition in which a number of key variables are not changing In the stochastic growth model, these key variables are for instance the growth. .. 2010 All rights reserved Even though the stochastic growth model itself might bear little resemblance to the real world, it has proven to be a useful framework that can easily be extended to account for a wide range of macroeconomic issues that are potentially important Kydland and Prescott (1982) introduced labor/leisure-substitution in the stochastic growth model, which gave rise to the so-called... ⎠ ∂ ln Ct ∗ (C.8) Appendix C The Stochastic Growth Model 1 1 + r∗ = Substituting in equation (C.8) gives then: Zt+1 = ˆ ˆ 1 − Ct+1 + Ct + rt+1 − r∗ 1 + r∗ Substituting in equation (C.6) and rearranging, gives then equation (22): ˆ Ct = rt+1 − r∗ ˆ Et Ct+1 − Et 1 + r∗ Download free eBooks at bookboon.com 29 Click on the ad to read more Appendix C The Stochastic Growth Model C6 The linearized equillibrium... eBooks at bookboon.com 31 Click on the ad to read more References The Stochastic Growth Model References Baxter, Marianne, and Robert G King (1993), ”Fiscal Policy in General Equilibrium”, American Economic Review 83 (June), 315-334 Campbell, John Y (1994), ”Inspecting the Mechanism: An Analytical Approach to the Stochastic Growth Model , Journal of Monetary Economics 33 (June), 463-506 Goodfriend,... balanced growth path 1+g = ∂ ln {(1 − δ)Kt + It } ∂It ∂It ∂ ln It = ϕ2 1 It (1 − δ)Kt + It = = 1 Kt+1 g+δ 1+g ∗ ∗ ∗ It ∗ ∗ as It /Kt = g + δ and Kt grows at rate g on the balanced growth path Substituting in equation (C.1) gives then the linearized law of motion for K: ˆ Kt+1 = g+δˆ 1−δ ˆ Kt + It 1+g 1+g .which is equation (19) Download free eBooks at bookboon.com 25 Appendix C The Stochastic Growth Model. .. it also violates the transversality conditions 11 Note that these values imply that the annual depreciation rate, the annual growth rate and the annual interest rate are about 10%, 2% and 6%, respectively Download free eBooks at bookboon.com 19 Appendix A The Stochastic Growth Model Appendix A A1 The maximization problem of the representative firm The maximization problem of the firm can be rewritten... ∗+δ r 1 1−α At L − G∗ t At L − G∗ t Now recall that on the balanced growth path, A and G grow at the rate of technological progress g The equation above then implies that C ∗ also grows at the rate g, such that our initial educated guess turns out to be correct Download free eBooks at bookboon.com 23 Appendix C The Stochastic Growth Model Appendix C C1 The linearized production function The production... at bookboon.com 24 Click on the ad to read more Appendix C The Stochastic Growth Model C2 The linearized law of motion of the capital stock The law of motion of the capital stock is given by equation (2): = Kt+1 (1 − δ)Kt + It Taking logarithms of both sides of this equation, and subtracting from both sides their values on the balanced growth path, yields: ∗ ln Kt+1 − ln Kt+1 ∗ = ln {(1 − δ)Kt + It...Solution of the linearized model The Stochastic Growth Model Now note that equation (31) is quadratic in ϕCK : Q0 + Q1 ϕCK + Q2 ϕ2 CK = 0 r ∗ +δ where Q0 = −(1 − α) 1+g , Q1 = (1 − α) (34) r ∗ +δ 1+r ∗ ∗ Ct ∗ (1+g) Kt − r ∗ −g 1+g and Q2 = ∗ . Koen Vermeylen The Stochastic Growth Model Downloadfreebooksat Download free eBooks at bookboon.com Click on the ad to read more The Stochastic Growth Model 2 Contents 1 a stic growth model. The stochastic growth model is a stochastic version of the neoclassical growth model with microfoundations, 1 and provides the backbone of a lot of macroeconomic models that. capital stock which is left after depreciation at time t +1. The stochastic growth model Download free eBooks at bookboon.com Click on the ad to read more The Stochastic Growth Model 5 The